The travelling salesman problem and adiabatic quantum computation: an algorithm

Abstract

An explicit algorithm for the travelling salesman problem is constructed in the framework of adiabatic quantum computation, AQC. The initial Hamiltonian for the AQC process admits canonical coherent states as the ground state, and the target Hamiltonian has the shortest tour as the desirable ground state. Some estimates/bounds are also given for the computational complexity of the algorithm with particular emphasis on the required energy resources, besides the space and time complexity, for the physical process of (quantum) computation in general.

In the paper [14], we obtain new class of time-energy uncertainty relations directly from the Schrödinger equation for time-dependent Hamiltonians in the general case. Our derivation as well as the results are new and different to those in the existing literature. It is important to note that only the initial state, neither the instantaneous eigenstates nor the full time-dependent wave function at any other times which would demand a full solution for a time-dependent Hamiltonian, is required for the time-energy relations. This hallmark of our results in [14] enables their wider applicability and usefulness.

In particular, we also obtain some results for the adiabatic quantum computation AQC with time-varying Hamiltonian \({\mathcal {H}}_G (t)\) in the time interval \(t\in [0,T]\) according to (1) and (3),

We can set, without loss of generality, the initial ground state energy to zero to obtain the various necessary conditions for the computing time \(T^{\mathrm{AQC}}\) at the end of the computation:

The necessary conditions above can also be expressed differently but equivalently as that the system cannot fully explore the whole Hilbert space (that is, cannot reach certain dynamically allowable state) or evolve into an orthogonal state from the initial state if the evolution time is less than, respectively, the following AQC characteristic times:

That is, if the computation time is less than \({\mathcal {T}}^{\mathrm{AQC}}_{\forall }\), then there exists some state which is allowed by the dynamics but cannot be reached from the initial state. And for evolution time less than \({\mathcal {T}}_{\mathrm{AQC}\perp }\), the system cannot evolve to any state that is orthogonal to the initial state.

The characteristic time in (A4) could be considered as an estimate of the lower bound on the computing time; as such, the more the energy and the more the spread of the initial state in energy with respect to the final Hamiltonian, the less the lower bound on computing time. Note also that the inverse of these characteristic times are related to the measures of the interpolation rates of the AQC Hamiltonian (1); the slower the rates, the higher the probabilities of ending the computation in the ground state of \(H_P\).

Our characteristic computing time estimates for AQC depend only on the initial state of the computation, its energy expectation and also its energy spread as measured in terms of the final (observable) Hamiltonian of the computation. These estimates are not explicitly but only implicitly dependent on the instantaneous energy gaps at intermediate times of the spectral flow of the AQC time-dependent Hamiltonian.

Appendix B: Energy resource as a component of computational complexity

We illustrate below the need for energy resources, not only in quantum computation but also in any physical computation, as an essential component for computational complexity, besides the usual resources of memory space and computing time. We illustrate this point with the aid of the following unstructured search algorithms in AQC.

We first consider a quantum adiabatic algorithm [17, 18] to locate the state \(|m\rangle \) in a unsorted database set of normalized orthogonal states \(\{|i\rangle , i = 1, \ldots , M\}\). It is known that this algorithm has a computational complexity of \({\mathcal {O}}(\sqrt{M})\) as that of Grover’s algorithm [19], a quadratic improvement on classical search.

For a AQC algorithm, we start with an initial state \(|\phi _0\rangle \) that is a uniform superposition of all the states in the given search set,

is the lower limit below which the initial state \(|\phi _0\rangle \)cannot evolve into an orthogonal state under the dynamics governed by \({\mathcal {H}}_G(t)\) in (1). This time limit is a typical measure of the computation time and thus should be of the same order of magnitude as the best AQC computation time, as estimated according to the quantum adiabatic theorem, to obtain the target state \(|m\rangle \) with reasonable probability.

For the case of the initial state is a uniform superposition of all the states, that is,

This time estimate, with g for which \( \int _0^1 g(\tau ) \mathrm{d}\tau \sim {\mathcal {O}}(1)\), is indeed of the same order of magnitude as the time complexity \({\mathcal {O}}(\root \of {M})\) for the AQC [17] as normally obtained from the energy gap of the two lowest eigenvalues in the spectral flow of \({\mathcal {H}}_G(t)\) according to the quantum adiabatic theorem.

In contrast to those derived from the quantum adiabatic theorem, the time estimate \(T_\perp ^{\mathrm{search}}\) here depends only on the extrapolating function g, the initial state and the target Hamiltonian. Our lower bound estimate, furthermore for this particular algorithm, is independent of all other amplitudes \(c_i\) for \(i\not = m\). It depends only on the coefficient \(c_m\) of the target state in the superposition (B1). We thus could improve on the time \({\mathcal {O}}(\sqrt{M})\) if we have some information that leads to higher priori probability for the target state \(|m\rangle \), such that \(|c_m| > 1/\sqrt{M}\).

In addition to that, we could also exploit the extra degree of freedom of the extrapolating function g to reduce the time estimate (B9). For example, with the choice

All of the above reductions for \(T_\perp ^{\mathrm{search}}\) match, in orders of magnitude, the time complexities derived in [16, 18] directly from a consideration of spectral-flow energy gap according to the quantum adiabatic theorem. This agreement is remarkable, as our results above are not derived directly from the quantum adiabatic theorem but from a general consideration of time-energy uncertainty relation for time-dependent Hamiltonians [14].

Such an agreement, however, is not unexpected. It should be reminded again here that our time measure \(T_\perp ^{\mathrm{search}}\) is a necessary lower limit in the sense that if the computation time is less than that then the initial state cannot evolve into an orthogonal state. But longer computation time, \(T > T_\perp ^{\mathrm{search}}\), is not a sufficient condition; for sufficiency, we would need to involve the quantum adiabatic theorem. However, the simply calculated time measure \(T_\perp ^{\mathrm{search}}\) does agree in order of magnitudes with the estimate of the computational time more comprehensively derived. This agreement of our results and those in [18] demonstrate that these necessary lower limits can in fact be saturated in this case by judicious choice of the extrapolating functions f and g [17, 18].

More importantly, we want to point out and emphasise here that although we may be able to reduce the time complexity to \({\mathcal {O}}(1)\), as with the choice of (B10)or (B11), we need to consider also the energy resource required for the computation. The choice of (B10) can, in fact, only be had at the cost of an increase in the energy required:

That is, a reduction in time complexity (from \({\mathcal {O}}(\sqrt{M})\) to \({\mathcal {O}}(1)\)) incurs and is balanced by an increase in the cost in energy resource (from \({\mathcal {O}}(1)\) to \({\mathcal {O}}(\sqrt{M})\)). That is, in general the computational complexity of the AQC (B4) is of the order \({\mathcal {O}}(\sqrt{M})\), taking into account both the energy and time resources.

The message here is that in considering the computational complexity in general we need to consider also the energy resources in addition to the usually considered space and time complexity.

See [20, 21, 22, 23, 24] for other AQC Hamiltonians for this search problem. For these and more general time-dependent Hamiltonians \(H(\tau )\), which do not assume the particular form of (1) and (3), we still could estimate some lower bounds on the computational time through the general uncertainty relations derived in [14]: